Digital Signal Processing Chapter 7: Parametric Spectrum Estimation

7.4.2 The Principle of OrthogonalityExamining the conjugate crosscorrelation-vector⎧⎡⎪⎨¯r EX = E{E ∗ (k) · X(k − )} = E⎢⎣⎪⎩and recalling thatE ∗ (k) · X(k − 1)E ∗ (k) · X(k − 2).E ∗ (k) · X(k − n)⎤⎫⎪⎬⎥⎦⎪⎭e(k) = x(k) − ¯p H x(k − ) ; e ∗ (k) = x ∗ (k) − x H (k − )¯p⇒ ¯r EX = E{[ X ∗ (k) − X H (k − ) ¯p ] · X(k − )}⇒ ¯r EX = E{X(k − ) · X ∗ (k)} − E{X(k − ) · X H (k − )} · ¯pExpressing the expectation values by the help of the autocorrelation-vector and -matrixwith further conjugation givesr EX = r XX − R XX p .Linear Prediction Page 9